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2: First Order Differential Equations

Last updated

Sep 5, 2021
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- 1: ODE Fundamentals
- Front Matter

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2.1: Difference Equations
Differential equation are great for modeling situations where there is a continually changing population or value. If the change happens incrementally rather than continuously then differential equations have their shortcomings. Instead we will use difference equations which are recursively defined sequences.
2.2: Classification of Differential Equations
Recall that a differential equation is an equation (has an equal sign) that involves derivatives. Just as biologists have a classification system for life, mathematicians have a classification system for differential equations. We can place all differential equation into two types: ordinary differential equation and partial differential equations.
2.3: Modeling with First Order Differential Equations
Whenever there is a process to be investigated, a mathematical model becomes a possibility. Since most processes involve something changing, derivatives come into play resulting in a differential equation. We will investigate examples of how differential equations can model such processes.
2.4: Separable Differential Equations
A differential equation is called separable if it can be written as f(y)dy=g(x)dx
2.5: Autonomous Differential Equations
A differential equation is called autonomous if it can be written as y'(t)=f(y). Autonomous differential equations are separable and can be solved by simple integration.
2.6: First Order Linear Differential Equations
In this section we will concentrate on first order linear differential equations. Recall that this means that only a first derivative appears in the differential equation and that the equation is linear.
2.7: Exact Differential Equations
That is if a differential equation can be written in a specific form, then we can seek the original function f(x,y) (called a potential function). A differential equation with a potential function is called exact. If you have had vector calculus, this is the same as finding the potential functions and using the fundamental theorem of line integrals.
2.8: Theory of Existence and Uniqueness
If a first order differential satisfies continuity conditions, then the initial value problem will have a unique solution in some neighborhood of the initial value.
2.9: Theory of Linear vs. Nonlinear Differential Equations
There exists a solution to all first order linear differential equations.

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